--- a/src/HOL/Library/RBT.thy Wed Mar 03 10:06:12 2010 +0100
+++ b/src/HOL/Library/RBT.thy Wed Mar 03 17:21:45 2010 +0100
@@ -10,6 +10,8 @@
imports Main AssocList
begin
+subsection {* Datatype of RB trees *}
+
datatype color = R | B
datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
@@ -23,39 +25,48 @@
case (Branch c) with that show thesis by (cases c) blast+
qed
-text {* Content of a tree *}
+subsection {* Tree properties *}
-primrec entries
+subsubsection {* Content of a tree *}
+
+primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
where
"entries Empty = []"
| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
-text {* Search tree properties *}
-
-primrec entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
+abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
where
- "entry_in_tree k v Empty = False"
-| "entry_in_tree k v (Branch c l x y r) \<longleftrightarrow> k = x \<and> v = y \<or> entry_in_tree k v l \<or> entry_in_tree k v r"
+ "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"
+
+definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where
+ "keys t = map fst (entries t)"
-primrec keys :: "('k, 'v) rbt \<Rightarrow> 'k set"
-where
- "keys Empty = {}"
-| "keys (Branch _ l k _ r) = { k } \<union> keys l \<union> keys r"
+lemma keys_simps [simp, code]:
+ "keys Empty = []"
+ "keys (Branch c l k v r) = keys l @ k # keys r"
+ by (simp_all add: keys_def)
lemma entry_in_tree_keys:
- "entry_in_tree k v t \<Longrightarrow> k \<in> keys t"
- by (induct t) auto
+ assumes "(k, v) \<in> set (entries t)"
+ shows "k \<in> set (keys t)"
+proof -
+ from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)
+ then show ?thesis by (simp add: keys_def)
+qed
+
+
+subsubsection {* Search tree properties *}
definition tree_less :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
where
- tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>keys t. x < k)"
+ tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>set (keys t). x < k)"
abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
where "t |\<guillemotleft> x \<equiv> tree_less x t"
definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50)
where
- tree_greater_prop: "tree_greater k t = (\<forall>x\<in>keys t. k < x)"
+ tree_greater_prop: "tree_greater k t = (\<forall>x\<in>set (keys t). k < x)"
lemma tree_less_simps [simp]:
"tree_less k Empty = True"
@@ -72,55 +83,172 @@
lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
-lemma tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+lemma tree_less_eq_trans: "l |\<guillemotleft> u \<Longrightarrow> u \<le> v \<Longrightarrow> l |\<guillemotleft> v"
+ and tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+ and tree_greater_eq_trans: "u \<le> v \<Longrightarrow> v \<guillemotleft>| r \<Longrightarrow> u \<guillemotleft>| r"
and tree_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
-by (auto simp: tree_ord_props)
+ by (auto simp: tree_ord_props)
primrec sorted :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
where
"sorted Empty = True"
| "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
+lemma sorted_entries:
+ "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
+by (induct t)
+ (force simp: sorted_append sorted_Cons tree_ord_props
+ dest!: entry_in_tree_keys)+
+
+lemma distinct_entries:
+ "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
+by (induct t)
+ (force simp: sorted_append sorted_Cons tree_ord_props
+ dest!: entry_in_tree_keys)+
+
+
+subsubsection {* Tree lookup *}
+
primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
where
"lookup Empty k = None"
| "lookup (Branch _ l x y r) k = (if k < x then lookup l k else if x < k then lookup r k else Some y)"
+lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = set (keys t)"
+ by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
+
+lemma dom_lookup_Branch:
+ "sorted (Branch c t1 k v t2) \<Longrightarrow>
+ dom (lookup (Branch c t1 k v t2))
+ = Set.insert k (dom (lookup t1) \<union> dom (lookup t2))"
+proof -
+ assume "sorted (Branch c t1 k v t2)"
+ moreover from this have "sorted t1" "sorted t2" by simp_all
+ ultimately show ?thesis by (simp add: lookup_keys)
+qed
+
+lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))"
+proof (induct t)
+ case Empty then show ?case by simp
+next
+ case (Branch color t1 a b t2)
+ let ?A = "Set.insert a (dom (lookup t1) \<union> dom (lookup t2))"
+ have "dom (lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
+ moreover from Branch have "finite (insert a (dom (lookup t1) \<union> dom (lookup t2)))" by simp
+ ultimately show ?case by (rule finite_subset)
+qed
+
lemma lookup_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None"
by (induct t) auto
lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
by (induct t) auto
-lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = keys t"
-by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
-
-lemma lookup_pit: "sorted t \<Longrightarrow> (lookup t k = Some v) = entry_in_tree k v t"
+lemma lookup_in_tree: "sorted t \<Longrightarrow> (lookup t k = Some v) = entry_in_tree k v t"
by (induct t) (auto simp: tree_less_prop tree_greater_prop entry_in_tree_keys)
lemma lookup_Empty: "lookup Empty = empty"
by (rule ext) simp
+lemma lookup_map_of_entries:
+ "sorted t \<Longrightarrow> lookup t = map_of (entries t)"
+proof (induct t)
+ case Empty thus ?case by (simp add: lookup_Empty)
+next
+ case (Branch c t1 k v t2)
+ have "lookup (Branch c t1 k v t2) = lookup t2 ++ [k\<mapsto>v] ++ lookup t1"
+ proof (rule ext)
+ fix x
+ from Branch have SORTED: "sorted (Branch c t1 k v t2)" by simp
+ let ?thesis = "lookup (Branch c t1 k v t2) x = (lookup t2 ++ [k \<mapsto> v] ++ lookup t1) x"
+
+ have DOM_T1: "!!k'. k'\<in>dom (lookup t1) \<Longrightarrow> k>k'"
+ proof -
+ fix k'
+ from SORTED have "t1 |\<guillemotleft> k" by simp
+ with tree_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
+ moreover assume "k'\<in>dom (lookup t1)"
+ ultimately show "k>k'" using lookup_keys SORTED by auto
+ qed
+
+ have DOM_T2: "!!k'. k'\<in>dom (lookup t2) \<Longrightarrow> k<k'"
+ proof -
+ fix k'
+ from SORTED have "k \<guillemotleft>| t2" by simp
+ with tree_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
+ moreover assume "k'\<in>dom (lookup t2)"
+ ultimately show "k<k'" using lookup_keys SORTED by auto
+ qed
+
+ {
+ assume C: "x<k"
+ hence "lookup (Branch c t1 k v t2) x = lookup t1 x" by simp
+ moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+ moreover have "x\<notin>dom (lookup t2)" proof
+ assume "x\<in>dom (lookup t2)"
+ with DOM_T2 have "k<x" by blast
+ with C show False by simp
+ qed
+ ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+ } moreover {
+ assume [simp]: "x=k"
+ hence "lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
+ moreover have "x\<notin>dom (lookup t1)" proof
+ assume "x\<in>dom (lookup t1)"
+ with DOM_T1 have "k>x" by blast
+ thus False by simp
+ qed
+ ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+ } moreover {
+ assume C: "x>k"
+ hence "lookup (Branch c t1 k v t2) x = lookup t2 x" by (simp add: less_not_sym[of k x])
+ moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
+ moreover have "x\<notin>dom (lookup t1)" proof
+ assume "x\<in>dom (lookup t1)"
+ with DOM_T1 have "k>x" by simp
+ with C show False by simp
+ qed
+ ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
+ } ultimately show ?thesis using less_linear by blast
+ qed
+ also from Branch have "lookup t2 ++ [k \<mapsto> v] ++ lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
+ finally show ?case .
+qed
+
+(*lemma map_of_inject:
+ assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
+ shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys"
+
+lemma entries_eqI:
+ assumes sorted: "sorted t1" "sorted t2"
+ assumes lookup: "lookup t1 = lookup t2"
+ shows entries: "entries t1 = entries t2"
+proof -
+ from sorted lookup have "map_of (entries t1) = map_of (entries t2)"
+ by (simp_all add: lookup_map_of_entries)
+qed*)
+
(* a kind of extensionality *)
-lemma lookup_from_pit:
+lemma lookup_from_in_tree:
assumes sorted: "sorted t1" "sorted t2"
and eq: "\<And>v. entry_in_tree (k\<Colon>'a\<Colon>linorder) v t1 = entry_in_tree k v t2"
shows "lookup t1 k = lookup t2 k"
proof (cases "lookup t1 k")
case None
then have "\<And>v. \<not> entry_in_tree k v t1"
- by (simp add: lookup_pit[symmetric] sorted)
+ by (simp add: lookup_in_tree[symmetric] sorted)
with None show ?thesis
- by (cases "lookup t2 k") (auto simp: lookup_pit sorted eq)
+ by (cases "lookup t2 k") (auto simp: lookup_in_tree sorted eq)
next
case (Some a)
then show ?thesis
apply (cases "lookup t2 k")
- apply (auto simp: lookup_pit sorted eq)
- by (auto simp add: lookup_pit[symmetric] sorted Some)
+ apply (auto simp: lookup_in_tree sorted eq)
+ by (auto simp add: lookup_in_tree[symmetric] sorted Some)
qed
-subsection {* Red-black properties *}
+
+subsubsection {* Red-black properties *}
primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
where
@@ -240,19 +368,23 @@
with 1 "5_4" show ?case by simp
qed simp+
-lemma keys_balance[simp]:
- "keys (balance l k v r) = { k } \<union> keys l \<union> keys r"
-by (induct l k v r rule: balance.induct) auto
+lemma entries_balance [simp]:
+ "entries (balance l k v r) = entries l @ (k, v) # entries r"
+ by (induct l k v r rule: balance.induct) auto
-lemma balance_pit:
- "entry_in_tree k x (balance l v y r) = (entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r)"
-by (induct l v y r rule: balance.induct) auto
+lemma keys_balance [simp]:
+ "keys (balance l k v r) = keys l @ k # keys r"
+ by (simp add: keys_def)
+
+lemma balance_in_tree:
+ "entry_in_tree k x (balance l v y r) \<longleftrightarrow> entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r"
+ by (auto simp add: keys_def)
lemma lookup_balance[simp]:
fixes k :: "'a::linorder"
assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
shows "lookup (balance l k v r) x = lookup (Branch B l k v r) x"
-by (rule lookup_from_pit) (auto simp:assms balance_pit balance_sorted)
+by (rule lookup_from_in_tree) (auto simp:assms balance_in_tree balance_sorted)
primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
@@ -264,7 +396,7 @@
lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
lemma paint_sorted[simp]: "sorted t \<Longrightarrow> sorted (paint c t)" by (cases t) auto
-lemma paint_pit[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
+lemma paint_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
lemma paint_lookup[simp]: "lookup (paint c t) = lookup t" by (rule ext) (cases t, auto)
lemma paint_tree_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
lemma paint_tree_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
@@ -294,8 +426,8 @@
lemma ins_sorted[simp]: "sorted t \<Longrightarrow> sorted (ins f k x t)"
by (induct f k x t rule: ins.induct) (auto simp: balance_sorted)
-lemma keys_ins: "keys (ins f k v t) = { k } \<union> keys t"
-by (induct f k v t rule: ins.induct) auto
+lemma keys_ins: "set (keys (ins f k v t)) = { k } \<union> set (keys t)"
+ by (induct f k v t rule: ins.induct) auto
lemma lookup_ins:
fixes k :: "'a::linorder"
@@ -305,45 +437,45 @@
using assms by (induct f k v t rule: ins.induct) auto
definition
- insertwithkey :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ insert_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "insertwithkey f k v t = paint B (ins f k v t)"
+ "insert_with_key f k v t = paint B (ins f k v t)"
-lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insertwithkey f k x t)"
- by (auto simp: insertwithkey_def)
+lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insert_with_key f k x t)"
+ by (auto simp: insert_with_key_def)
theorem insertwk_is_rbt:
assumes inv: "is_rbt t"
- shows "is_rbt (insertwithkey f k x t)"
+ shows "is_rbt (insert_with_key f k x t)"
using assms
-unfolding insertwithkey_def is_rbt_def
+unfolding insert_with_key_def is_rbt_def
by (auto simp: ins_inv1_inv2)
lemma lookup_insertwk:
assumes "sorted t"
- shows "lookup (insertwithkey f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v
+ shows "lookup (insert_with_key f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v
| Some w \<Rightarrow> f k w v)) x"
-unfolding insertwithkey_def using assms
+unfolding insert_with_key_def using assms
by (simp add:lookup_ins)
definition
- insertw_def: "insertwith f = insertwithkey (\<lambda>_. f)"
+ insertw_def: "insert_with f = insert_with_key (\<lambda>_. f)"
-lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insertwith f k v t)" by (simp add: insertwk_sorted insertw_def)
-theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insertwith f k v t)" by (simp add: insertwk_is_rbt insertw_def)
+lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insert_with f k v t)" by (simp add: insertwk_sorted insertw_def)
+theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert_with f k v t)" by (simp add: insertwk_is_rbt insertw_def)
lemma lookup_insertw:
assumes "is_rbt t"
- shows "lookup (insertwith f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
+ shows "lookup (insert_with f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
using assms
unfolding insertw_def
by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
- "insert k v t = insertwithkey (\<lambda>_ _ nv. nv) k v t"
+ "insert = insert_with_key (\<lambda>_ _ nv. nv)"
lemma insert_sorted: "sorted t \<Longrightarrow> sorted (insert k v t)" by (simp add: insertwk_sorted insert_def)
-theorem insert_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
+theorem insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
lemma lookup_insert:
assumes "is_rbt t"
@@ -359,178 +491,174 @@
by (cases t rule: rbt_cases) auto
fun
- balleft :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "balleft (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
- "balleft bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
- "balleft bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
- "balleft t k x s = Empty"
+ "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
+ "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
+ "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
+ "balance_left t k x s = Empty"
-lemma balleft_inv2_with_inv1:
+lemma balance_left_inv2_with_inv1:
assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
- shows "bheight (balleft lt k v rt) = bheight lt + 1"
- and "inv2 (balleft lt k v rt)"
+ shows "bheight (balance_left lt k v rt) = bheight lt + 1"
+ and "inv2 (balance_left lt k v rt)"
using assms
-by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bheight)
+by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)
-lemma balleft_inv2_app:
+lemma balance_left_inv2_app:
assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
- shows "inv2 (balleft lt k v rt)"
- "bheight (balleft lt k v rt) = bheight rt"
+ shows "inv2 (balance_left lt k v rt)"
+ "bheight (balance_left lt k v rt) = bheight rt"
using assms
-by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bheight)+
+by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+
-lemma balleft_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balleft a k x b)"
- by (induct a k x b rule: balleft.induct) (simp add: balance_inv1)+
+lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"
+ by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+
-lemma balleft_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balleft lt k x rt)"
-by (induct lt k x rt rule: balleft.induct) (auto simp: balance_inv1)
+lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"
+by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)
-lemma balleft_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balleft l k v r)"
-apply (induct l k v r rule: balleft.induct)
+lemma balance_left_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_left l k v r)"
+apply (induct l k v r rule: balance_left.induct)
apply (auto simp: balance_sorted)
apply (unfold tree_greater_prop tree_less_prop)
by force+
-lemma balleft_tree_greater:
+lemma balance_left_tree_greater:
fixes k :: "'a::order"
assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x"
- shows "k \<guillemotleft>| balleft a x t b"
+ shows "k \<guillemotleft>| balance_left a x t b"
using assms
-by (induct a x t b rule: balleft.induct) auto
+by (induct a x t b rule: balance_left.induct) auto
-lemma balleft_tree_less:
+lemma balance_left_tree_less:
fixes k :: "'a::order"
assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k"
- shows "balleft a x t b |\<guillemotleft> k"
+ shows "balance_left a x t b |\<guillemotleft> k"
using assms
-by (induct a x t b rule: balleft.induct) auto
+by (induct a x t b rule: balance_left.induct) auto
-lemma balleft_pit:
+lemma balance_left_in_tree:
assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight r"
- shows "entry_in_tree k v (balleft l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
+ shows "entry_in_tree k v (balance_left l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
using assms
-by (induct l k v r rule: balleft.induct) (auto simp: balance_pit)
+by (induct l k v r rule: balance_left.induct) (auto simp: balance_in_tree)
fun
- balright :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ balance_right :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "balright a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
- "balright (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
- "balright (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
- "balright t k x s = Empty"
+ "balance_right a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
+ "balance_right (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
+ "balance_right (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
+ "balance_right t k x s = Empty"
-lemma balright_inv2_with_inv1:
+lemma balance_right_inv2_with_inv1:
assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
- shows "inv2 (balright lt k v rt) \<and> bheight (balright lt k v rt) = bheight lt"
+ shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
using assms
-by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bheight)
+by (induct lt k v rt rule: balance_right.induct) (auto simp: balance_inv2 balance_bheight)
-lemma balright_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balright a k x b)"
-by (induct a k x b rule: balright.induct) (simp add: balance_inv1)+
+lemma balance_right_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balance_right a k x b)"
+by (induct a k x b rule: balance_right.induct) (simp add: balance_inv1)+
-lemma balright_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balright lt k x rt)"
-by (induct lt k x rt rule: balright.induct) (auto simp: balance_inv1)
+lemma balance_right_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balance_right lt k x rt)"
+by (induct lt k x rt rule: balance_right.induct) (auto simp: balance_inv1)
-lemma balright_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balright l k v r)"
-apply (induct l k v r rule: balright.induct)
+lemma balance_right_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balance_right l k v r)"
+apply (induct l k v r rule: balance_right.induct)
apply (auto simp:balance_sorted)
apply (unfold tree_less_prop tree_greater_prop)
by force+
-lemma balright_tree_greater:
+lemma balance_right_tree_greater:
fixes k :: "'a::order"
assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x"
- shows "k \<guillemotleft>| balright a x t b"
-using assms by (induct a x t b rule: balright.induct) auto
+ shows "k \<guillemotleft>| balance_right a x t b"
+using assms by (induct a x t b rule: balance_right.induct) auto
-lemma balright_tree_less:
+lemma balance_right_tree_less:
fixes k :: "'a::order"
assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k"
- shows "balright a x t b |\<guillemotleft> k"
-using assms by (induct a x t b rule: balright.induct) auto
+ shows "balance_right a x t b |\<guillemotleft> k"
+using assms by (induct a x t b rule: balance_right.induct) auto
-lemma balright_pit:
+lemma balance_right_in_tree:
assumes "inv1 l" "inv1l r" "bheight l = bheight r + 1" "inv2 l" "inv2 r"
- shows "entry_in_tree x y (balright l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
-using assms by (induct l k v r rule: balright.induct) (auto simp: balance_pit)
-
-
-text {* app *}
+ shows "entry_in_tree x y (balance_right l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
+using assms by (induct l k v r rule: balance_right.induct) (auto simp: balance_in_tree)
fun
- app :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ combine :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "app Empty x = x"
-| "app x Empty = x"
-| "app (Branch R a k x b) (Branch R c s y d) = (case (app b c) of
+ "combine Empty x = x"
+| "combine x Empty = x"
+| "combine (Branch R a k x b) (Branch R c s y d) = (case (combine b c) of
Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
bc \<Rightarrow> Branch R a k x (Branch R bc s y d))"
-| "app (Branch B a k x b) (Branch B c s y d) = (case (app b c) of
+| "combine (Branch B a k x b) (Branch B c s y d) = (case (combine b c) of
Branch R b2 t z c2 \<Rightarrow> Branch R (Branch B a k x b2) t z (Branch B c2 s y d) |
- bc \<Rightarrow> balleft a k x (Branch B bc s y d))"
-| "app a (Branch R b k x c) = Branch R (app a b) k x c"
-| "app (Branch R a k x b) c = Branch R a k x (app b c)"
+ bc \<Rightarrow> balance_left a k x (Branch B bc s y d))"
+| "combine a (Branch R b k x c) = Branch R (combine a b) k x c"
+| "combine (Branch R a k x b) c = Branch R a k x (combine b c)"
-lemma app_inv2:
+lemma combine_inv2:
assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
- shows "bheight (app lt rt) = bheight lt" "inv2 (app lt rt)"
+ shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
using assms
-by (induct lt rt rule: app.induct)
- (auto simp: balleft_inv2_app split: rbt.splits color.splits)
+by (induct lt rt rule: combine.induct)
+ (auto simp: balance_left_inv2_app split: rbt.splits color.splits)
-lemma app_inv1:
+lemma combine_inv1:
assumes "inv1 lt" "inv1 rt"
- shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (app lt rt)"
- "inv1l (app lt rt)"
+ shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (combine lt rt)"
+ "inv1l (combine lt rt)"
using assms
-by (induct lt rt rule: app.induct)
- (auto simp: balleft_inv1 split: rbt.splits color.splits)
+by (induct lt rt rule: combine.induct)
+ (auto simp: balance_left_inv1 split: rbt.splits color.splits)
-lemma app_tree_greater[simp]:
+lemma combine_tree_greater[simp]:
fixes k :: "'a::linorder"
assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r"
- shows "k \<guillemotleft>| app l r"
+ shows "k \<guillemotleft>| combine l r"
using assms
-by (induct l r rule: app.induct)
- (auto simp: balleft_tree_greater split:rbt.splits color.splits)
+by (induct l r rule: combine.induct)
+ (auto simp: balance_left_tree_greater split:rbt.splits color.splits)
-lemma app_tree_less[simp]:
+lemma combine_tree_less[simp]:
fixes k :: "'a::linorder"
assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k"
- shows "app l r |\<guillemotleft> k"
+ shows "combine l r |\<guillemotleft> k"
using assms
-by (induct l r rule: app.induct)
- (auto simp: balleft_tree_less split:rbt.splits color.splits)
+by (induct l r rule: combine.induct)
+ (auto simp: balance_left_tree_less split:rbt.splits color.splits)
-lemma app_sorted:
+lemma combine_sorted:
fixes k :: "'a::linorder"
assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
- shows "sorted (app l r)"
-using assms proof (induct l r rule: app.induct)
+ shows "sorted (combine l r)"
+using assms proof (induct l r rule: combine.induct)
case (3 a x v b c y w d)
hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
by auto
with 3
show ?case
- apply (cases "app b c" rule: rbt_cases)
- apply auto
- by (metis app_tree_greater app_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+
+ by (cases "combine b c" rule: rbt_cases)
+ (auto, (metis combine_tree_greater combine_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+)
next
case (4 a x v b c y w d)
hence "x < k \<and> tree_greater k c" by simp
hence "tree_greater x c" by (blast dest: tree_greater_trans)
- with 4 have 2: "tree_greater x (app b c)" by (simp add: app_tree_greater)
+ with 4 have 2: "tree_greater x (combine b c)" by (simp add: combine_tree_greater)
from 4 have "k < y \<and> tree_less k b" by simp
hence "tree_less y b" by (blast dest: tree_less_trans)
- with 4 have 3: "tree_less y (app b c)" by (simp add: app_tree_less)
+ with 4 have 3: "tree_less y (combine b c)" by (simp add: combine_tree_less)
show ?case
- proof (cases "app b c" rule: rbt_cases)
+ proof (cases "combine b c" rule: rbt_cases)
case Empty
from 4 have "x < y \<and> tree_greater y d" by auto
hence "tree_greater x d" by (blast dest: tree_greater_trans)
with 4 Empty have "sorted a" and "sorted (Branch B Empty y w d)" and "tree_less x a" and "tree_greater x (Branch B Empty y w d)" by auto
- with Empty show ?thesis by (simp add: balleft_sorted)
+ with Empty show ?thesis by (simp add: balance_left_sorted)
next
case (Red lta va ka rta)
with 2 4 have "x < va \<and> tree_less x a" by simp
@@ -542,71 +670,71 @@
case (Black lta va ka rta)
from 4 have "x < y \<and> tree_greater y d" by auto
hence "tree_greater x d" by (blast dest: tree_greater_trans)
- with Black 2 3 4 have "sorted a" and "sorted (Branch B (app b c) y w d)" and "tree_less x a" and "tree_greater x (Branch B (app b c) y w d)" by auto
- with Black show ?thesis by (simp add: balleft_sorted)
+ with Black 2 3 4 have "sorted a" and "sorted (Branch B (combine b c) y w d)" and "tree_less x a" and "tree_greater x (Branch B (combine b c) y w d)" by auto
+ with Black show ?thesis by (simp add: balance_left_sorted)
qed
next
case (5 va vb vd vc b x w c)
hence "k < x \<and> tree_less k (Branch B va vb vd vc)" by simp
hence "tree_less x (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
- with 5 show ?case by (simp add: app_tree_less)
+ with 5 show ?case by (simp add: combine_tree_less)
next
case (6 a x v b va vb vd vc)
hence "x < k \<and> tree_greater k (Branch B va vb vd vc)" by simp
hence "tree_greater x (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
- with 6 show ?case by (simp add: app_tree_greater)
+ with 6 show ?case by (simp add: combine_tree_greater)
qed simp+
-lemma app_pit:
+lemma combine_in_tree:
assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
- shows "entry_in_tree k v (app l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
+ shows "entry_in_tree k v (combine l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
using assms
-proof (induct l r rule: app.induct)
+proof (induct l r rule: combine.induct)
case (4 _ _ _ b c)
- hence a: "bheight (app b c) = bheight b" by (simp add: app_inv2)
- from 4 have b: "inv1l (app b c)" by (simp add: app_inv1)
+ hence a: "bheight (combine b c) = bheight b" by (simp add: combine_inv2)
+ from 4 have b: "inv1l (combine b c)" by (simp add: combine_inv1)
show ?case
- proof (cases "app b c" rule: rbt_cases)
+ proof (cases "combine b c" rule: rbt_cases)
case Empty
- with 4 a show ?thesis by (auto simp: balleft_pit)
+ with 4 a show ?thesis by (auto simp: balance_left_in_tree)
next
case (Red lta ka va rta)
with 4 show ?thesis by auto
next
case (Black lta ka va rta)
- with a b 4 show ?thesis by (auto simp: balleft_pit)
+ with a b 4 show ?thesis by (auto simp: balance_left_in_tree)
qed
qed (auto split: rbt.splits color.splits)
fun
- delformLeft :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
- delformRight :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+ del_from_left :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
+ del_from_right :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
del :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
"del x Empty = Empty" |
- "del x (Branch c a y s b) = (if x < y then delformLeft x a y s b else (if x > y then delformRight x a y s b else app a b))" |
- "delformLeft x (Branch B lt z v rt) y s b = balleft (del x (Branch B lt z v rt)) y s b" |
- "delformLeft x a y s b = Branch R (del x a) y s b" |
- "delformRight x a y s (Branch B lt z v rt) = balright a y s (del x (Branch B lt z v rt))" |
- "delformRight x a y s b = Branch R a y s (del x b)"
+ "del x (Branch c a y s b) = (if x < y then del_from_left x a y s b else (if x > y then del_from_right x a y s b else combine a b))" |
+ "del_from_left x (Branch B lt z v rt) y s b = balance_left (del x (Branch B lt z v rt)) y s b" |
+ "del_from_left x a y s b = Branch R (del x a) y s b" |
+ "del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (del x (Branch B lt z v rt))" |
+ "del_from_right x a y s b = Branch R a y s (del x b)"
lemma
assumes "inv2 lt" "inv1 lt"
shows
"\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
- inv2 (delformLeft x lt k v rt) \<and> bheight (delformLeft x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformLeft x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformLeft x lt k v rt))"
+ inv2 (del_from_left x lt k v rt) \<and> bheight (del_from_left x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (del_from_left x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (del_from_left x lt k v rt))"
and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
- inv2 (delformRight x lt k v rt) \<and> bheight (delformRight x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformRight x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformRight x lt k v rt))"
+ inv2 (del_from_right x lt k v rt) \<and> bheight (del_from_right x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (del_from_right x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (del_from_right x lt k v rt))"
and del_inv1_inv2: "inv2 (del x lt) \<and> (color_of lt = R \<and> bheight (del x lt) = bheight lt \<and> inv1 (del x lt)
\<or> color_of lt = B \<and> bheight (del x lt) = bheight lt - 1 \<and> inv1l (del x lt))"
using assms
-proof (induct x lt k v rt and x lt k v rt and x lt rule: delformLeft_delformRight_del.induct)
+proof (induct x lt k v rt and x lt k v rt and x lt rule: del_from_left_del_from_right_del.induct)
case (2 y c _ y')
have "y = y' \<or> y < y' \<or> y > y'" by auto
thus ?case proof (elim disjE)
assume "y = y'"
- with 2 show ?thesis by (cases c) (simp add: app_inv2 app_inv1)+
+ with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
next
assume "y < y'"
with 2 show ?thesis by (cases c) auto
@@ -616,35 +744,35 @@
qed
next
case (3 y lt z v rta y' ss bb)
- thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balleft_inv2_with_inv1 balleft_inv1 balleft_inv1l)+
+ thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
next
case (5 y a y' ss lt z v rta)
- thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balright_inv2_with_inv1 balright_inv1 balright_inv1l)+
+ thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
next
case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
qed auto
lemma
- delformLeft_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformLeft x lt k y rt)"
- and delformRight_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformRight x lt k y rt)"
+ del_from_left_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (del_from_left x lt k y rt)"
+ and del_from_right_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (del_from_right x lt k y rt)"
and del_tree_less: "tree_less v lt \<Longrightarrow> tree_less v (del x lt)"
-by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
- (auto simp: balleft_tree_less balright_tree_less)
+by (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct)
+ (auto simp: balance_left_tree_less balance_right_tree_less)
-lemma delformLeft_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformLeft x lt k y rt)"
- and delformRight_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformRight x lt k y rt)"
+lemma del_from_left_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (del_from_left x lt k y rt)"
+ and del_from_right_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (del_from_right x lt k y rt)"
and del_tree_greater: "tree_greater v lt \<Longrightarrow> tree_greater v (del x lt)"
-by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
- (auto simp: balleft_tree_greater balright_tree_greater)
+by (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct)
+ (auto simp: balance_left_tree_greater balance_right_tree_greater)
-lemma "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformLeft x lt k y rt)"
- and "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformRight x lt k y rt)"
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (del_from_left x lt k y rt)"
+ and "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (del_from_right x lt k y rt)"
and del_sorted: "sorted lt \<Longrightarrow> sorted (del x lt)"
-proof (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
+proof (induct x lt k y rt and x lt k y rt and x lt rule: del_from_left_del_from_right_del.induct)
case (3 x lta zz v rta yy ss bb)
from 3 have "tree_less yy (Branch B lta zz v rta)" by simp
hence "tree_less yy (del x (Branch B lta zz v rta))" by (rule del_tree_less)
- with 3 show ?case by (simp add: balleft_sorted)
+ with 3 show ?case by (simp add: balance_left_sorted)
next
case ("4_2" x vaa vbb vdd vc yy ss bb)
hence "tree_less yy (Branch R vaa vbb vdd vc)" by simp
@@ -654,18 +782,18 @@
case (5 x aa yy ss lta zz v rta)
hence "tree_greater yy (Branch B lta zz v rta)" by simp
hence "tree_greater yy (del x (Branch B lta zz v rta))" by (rule del_tree_greater)
- with 5 show ?case by (simp add: balright_sorted)
+ with 5 show ?case by (simp add: balance_right_sorted)
next
case ("6_2" x aa yy ss vaa vbb vdd vc)
hence "tree_greater yy (Branch R vaa vbb vdd vc)" by simp
hence "tree_greater yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_greater)
with "6_2" show ?case by simp
-qed (auto simp: app_sorted)
+qed (auto simp: combine_sorted)
-lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformLeft x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
- and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformRight x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
- and del_pit: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
-proof (induct x lt kt y rt and x lt kt y rt and x t rule: delformLeft_delformRight_del.induct)
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+ and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+ and del_in_tree: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
+proof (induct x lt kt y rt and x lt kt y rt and x t rule: del_from_left_del_from_right_del.induct)
case (2 xx c aa yy ss bb)
have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
from this 2 show ?case proof (elim disjE)
@@ -674,15 +802,15 @@
case True
from 2 `xx = yy` `xx = k` have "sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: tree_less_nit tree_greater_prop)
- with `xx = yy` 2 `xx = k` show ?thesis by (simp add: app_pit)
- qed (simp add: app_pit)
+ with `xx = yy` 2 `xx = k` show ?thesis by (simp add: combine_in_tree)
+ qed (simp add: combine_in_tree)
qed simp+
next
case (3 xx lta zz vv rta yy ss bb)
def mt[simp]: mt == "Branch B lta zz vv rta"
from 3 have "inv2 mt \<and> inv1 mt" by simp
hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
- with 3 have 4: "entry_in_tree k v (delformLeft xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balleft_pit)
+ with 3 have 4: "entry_in_tree k v (del_from_left xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balance_left_in_tree)
thus ?case proof (cases "xx = k")
case True
from 3 True have "tree_greater yy bb \<and> yy > k" by simp
@@ -706,13 +834,13 @@
with "4_2" have "k < yy \<and> tree_greater yy bb" by simp
hence "tree_greater k bb" by (blast dest: tree_greater_trans)
with True "4_2" show ?thesis by (auto simp: tree_greater_nit)
- qed simp
+ qed auto
next
case (5 xx aa yy ss lta zz vv rta)
def mt[simp]: mt == "Branch B lta zz vv rta"
from 5 have "inv2 mt \<and> inv1 mt" by simp
hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
- with 5 have 3: "entry_in_tree k v (delformRight xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balright_pit)
+ with 5 have 3: "entry_in_tree k v (del_from_right xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balance_right_in_tree)
thus ?case proof (cases "xx = k")
case True
from 5 True have "tree_less yy aa \<and> yy < k" by simp
@@ -734,14 +862,14 @@
with "6_2" have "k > yy \<and> tree_less yy aa" by simp
hence "tree_less k aa" by (blast dest: tree_less_trans)
with True "6_2" show ?thesis by (auto simp: tree_less_nit)
- qed simp
+ qed auto
qed simp
definition delete where
delete_def: "delete k t = paint B (del k t)"
-theorem delete_is_rbt[simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
+theorem delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
proof -
from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto
hence "inv2 (del k t) \<and> (color_of t = R \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color_of t = B \<and> bheight (del k t) = bheight t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
@@ -751,11 +879,11 @@
by (auto intro: paint_sorted del_sorted)
qed
-lemma delete_pit:
+lemma delete_in_tree:
assumes "is_rbt t"
shows "entry_in_tree k v (delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
using assms unfolding is_rbt_def delete_def
- by (auto simp: del_pit)
+ by (auto simp: del_in_tree)
lemma lookup_delete:
assumes is_rbt: "is_rbt t"
@@ -766,35 +894,36 @@
proof (cases "x = k")
assume "x = k"
with is_rbt show ?thesis
- by (cases "lookup (delete k t) k") (auto simp: lookup_pit delete_pit)
+ by (cases "lookup (delete k t) k") (auto simp: lookup_in_tree delete_in_tree)
next
assume "x \<noteq> k"
thus ?thesis
- by auto (metis is_rbt delete_is_rbt delete_pit is_rbt_sorted lookup_from_pit)
+ by auto (metis is_rbt delete_is_rbt delete_in_tree is_rbt_sorted lookup_from_in_tree)
qed
qed
+
subsection {* Union *}
primrec
- unionwithkey :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ union_with_key :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "unionwithkey f t Empty = t"
-| "unionwithkey f t (Branch c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
+ "union_with_key f t Empty = t"
+| "union_with_key f t (Branch c lt k v rt) = union_with_key f (union_with_key f (insert_with_key f k v t) lt) rt"
-lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (unionwithkey f lt rt)"
+lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (union_with_key f lt rt)"
by (induct rt arbitrary: lt) (auto simp: insertwk_sorted)
-theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (unionwithkey f lt rt)"
+theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (union_with_key f lt rt)"
by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
definition
- unionwith where
- "unionwith f = unionwithkey (\<lambda>_. f)"
+ union_with where
+ "union_with f = union_with_key (\<lambda>_. f)"
-theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (unionwith f lt rt)" unfolding unionwith_def by simp
+theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union_with f lt rt)" unfolding union_with_def by simp
definition union where
- "union = unionwithkey (%_ _ rv. rv)"
+ "union = union_with_key (%_ _ rv. rv)"
theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
@@ -811,7 +940,7 @@
case Empty thus ?case by (auto simp: union_def)
next
case (Branch c l k v r s)
- hence sortedrl: "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+ then have "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
have meq: "lookup s(k \<mapsto> v) ++ lookup l ++ lookup r =
lookup s ++
@@ -839,178 +968,79 @@
qed
qed
- from Branch
- have IHs:
+ from Branch have is_rbt: "is_rbt (RBT.union (RBT.insert k v s) l)"
+ by (auto intro: union_is_rbt insert_is_rbt)
+ with Branch have IHs:
"lookup (union (union (insert k v s) l) r) = lookup (union (insert k v s) l) ++ lookup r"
"lookup (union (insert k v s) l) = lookup (insert k v s) ++ lookup l"
- by (auto intro: union_is_rbt insert_is_rbt)
+ by auto
with meq show ?case
by (auto simp: lookup_insert[OF Branch(3)])
+
qed
-subsection {* Adjust *}
+
+subsection {* Modifying existing entries *}
primrec
- adjustwithkey :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
+ map_entry :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "adjustwithkey f k Empty = Empty"
-| "adjustwithkey f k (Branch c lt x v rt) = (if k < x then (Branch c (adjustwithkey f k lt) x v rt) else if k > x then (Branch c lt x v (adjustwithkey f k rt)) else (Branch c lt x (f x v) rt))"
+ "map_entry f k Empty = Empty"
+| "map_entry f k (Branch c lt x v rt) = (if k < x then (Branch c (map_entry f k lt) x v rt) else if k > x then (Branch c lt x v (map_entry f k rt)) else (Branch c lt x (f x v) rt))"
-lemma adjustwk_color_of: "color_of (adjustwithkey f k t) = color_of t" by (induct t) simp+
-lemma adjustwk_inv1: "inv1 (adjustwithkey f k t) = inv1 t" by (induct t) (simp add: adjustwk_color_of)+
-lemma adjustwk_inv2: "inv2 (adjustwithkey f k t) = inv2 t" "bheight (adjustwithkey f k t) = bheight t" by (induct t) simp+
-lemma adjustwk_tree_greater: "tree_greater k (adjustwithkey f kk t) = tree_greater k t" by (induct t) simp+
-lemma adjustwk_tree_less: "tree_less k (adjustwithkey f kk t) = tree_less k t" by (induct t) simp+
-lemma adjustwk_sorted: "sorted (adjustwithkey f k t) = sorted t" by (induct t) (simp add: adjustwk_tree_less adjustwk_tree_greater)+
+lemma map_entrywk_color_of: "color_of (map_entry f k t) = color_of t" by (induct t) simp+
+lemma map_entrywk_inv1: "inv1 (map_entry f k t) = inv1 t" by (induct t) (simp add: map_entrywk_color_of)+
+lemma map_entrywk_inv2: "inv2 (map_entry f k t) = inv2 t" "bheight (map_entry f k t) = bheight t" by (induct t) simp+
+lemma map_entrywk_tree_greater: "tree_greater k (map_entry f kk t) = tree_greater k t" by (induct t) simp+
+lemma map_entrywk_tree_less: "tree_less k (map_entry f kk t) = tree_less k t" by (induct t) simp+
+lemma map_entrywk_sorted: "sorted (map_entry f k t) = sorted t" by (induct t) (simp add: map_entrywk_tree_less map_entrywk_tree_greater)+
-theorem adjustwk_is_rbt[simp]: "is_rbt (adjustwithkey f k t) = is_rbt t"
-unfolding is_rbt_def by (simp add: adjustwk_inv2 adjustwk_color_of adjustwk_sorted adjustwk_inv1 )
+theorem map_entrywk_is_rbt [simp]: "is_rbt (map_entry f k t) = is_rbt t"
+unfolding is_rbt_def by (simp add: map_entrywk_inv2 map_entrywk_color_of map_entrywk_sorted map_entrywk_inv1 )
-theorem adjustwithkey_map[simp]:
- "lookup (adjustwithkey f k t) x =
+theorem map_entry_map [simp]:
+ "lookup (map_entry f k t) x =
(if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f k y)
else lookup t x)"
by (induct t arbitrary: x) (auto split:option.splits)
-definition adjust where
- "adjust f = adjustwithkey (\<lambda>_. f)"
-theorem adjust_is_rbt[simp]: "is_rbt (adjust f k t) = is_rbt t" unfolding adjust_def by simp
-
-theorem adjust_map[simp]:
- "lookup (adjust f k t) x =
- (if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f y)
- else lookup t x)"
-unfolding adjust_def by simp
-
-subsection {* Map *}
-
-primrec
- mapwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
-where
- "mapwithkey f Empty = Empty"
-| "mapwithkey f (Branch c lt k v rt) = Branch c (mapwithkey f lt) k (f k v) (mapwithkey f rt)"
-
-theorem mapwk_keys[simp]: "keys (mapwithkey f t) = keys t" by (induct t) auto
-lemma mapwk_tree_greater: "tree_greater k (mapwithkey f t) = tree_greater k t" by (induct t) simp+
-lemma mapwk_tree_less: "tree_less k (mapwithkey f t) = tree_less k t" by (induct t) simp+
-lemma mapwk_sorted: "sorted (mapwithkey f t) = sorted t" by (induct t) (simp add: mapwk_tree_less mapwk_tree_greater)+
-lemma mapwk_color_of: "color_of (mapwithkey f t) = color_of t" by (induct t) simp+
-lemma mapwk_inv1: "inv1 (mapwithkey f t) = inv1 t" by (induct t) (simp add: mapwk_color_of)+
-lemma mapwk_inv2: "inv2 (mapwithkey f t) = inv2 t" "bheight (mapwithkey f t) = bheight t" by (induct t) simp+
-theorem mapwk_is_rbt[simp]: "is_rbt (mapwithkey f t) = is_rbt t"
-unfolding is_rbt_def by (simp add: mapwk_inv1 mapwk_inv2 mapwk_sorted mapwk_color_of)
-
-theorem lookup_mapwk[simp]: "lookup (mapwithkey f t) x = Option.map (f x) (lookup t x)"
-by (induct t) auto
-
-definition map
-where map_def: "map f == mapwithkey (\<lambda>_. f)"
-
-theorem map_keys[simp]: "keys (map f t) = keys t" unfolding map_def by simp
-theorem map_is_rbt[simp]: "is_rbt (map f t) = is_rbt t" unfolding map_def by simp
-theorem lookup_map[simp]: "lookup (map f t) = Option.map f o lookup t"
- by (rule ext) (simp add:map_def)
-
-subsection {* Fold *}
-
-text {* The following is still incomplete... *}
+subsection {* Mapping all entries *}
primrec
- foldwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
+ map :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
where
- "foldwithkey f Empty v = v"
-| "foldwithkey f (Branch c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
-
-lemma lookup_entries_aux: "sorted (Branch c t1 k v t2) \<Longrightarrow> RBT.lookup (Branch c t1 k v t2) = RBT.lookup t2 ++ [k\<mapsto>v] ++ RBT.lookup t1"
-proof (rule ext)
- fix x
- assume SORTED: "sorted (Branch c t1 k v t2)"
- let ?thesis = "RBT.lookup (Branch c t1 k v t2) x = (RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1) x"
-
- have DOM_T1: "!!k'. k'\<in>dom (RBT.lookup t1) \<Longrightarrow> k>k'"
- proof -
- fix k'
- from SORTED have "t1 |\<guillemotleft> k" by simp
- with tree_less_prop have "\<forall>k'\<in>keys t1. k>k'" by auto
- moreover assume "k'\<in>dom (RBT.lookup t1)"
- ultimately show "k>k'" using RBT.lookup_keys SORTED by auto
- qed
-
- have DOM_T2: "!!k'. k'\<in>dom (RBT.lookup t2) \<Longrightarrow> k<k'"
- proof -
- fix k'
- from SORTED have "k \<guillemotleft>| t2" by simp
- with tree_greater_prop have "\<forall>k'\<in>keys t2. k<k'" by auto
- moreover assume "k'\<in>dom (RBT.lookup t2)"
- ultimately show "k<k'" using RBT.lookup_keys SORTED by auto
- qed
+ "map f Empty = Empty"
+| "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"
- {
- assume C: "x<k"
- hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t1 x" by simp
- moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
- moreover have "x\<notin>dom (RBT.lookup t2)" proof
- assume "x\<in>dom (RBT.lookup t2)"
- with DOM_T2 have "k<x" by blast
- with C show False by simp
- qed
- ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
- } moreover {
- assume [simp]: "x=k"
- hence "RBT.lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
- moreover have "x\<notin>dom (RBT.lookup t1)" proof
- assume "x\<in>dom (RBT.lookup t1)"
- with DOM_T1 have "k>x" by blast
- thus False by simp
- qed
- ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
- } moreover {
- assume C: "x>k"
- hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t2 x" by (simp add: less_not_sym[of k x])
- moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
- moreover have "x\<notin>dom (RBT.lookup t1)" proof
- assume "x\<in>dom (RBT.lookup t1)"
- with DOM_T1 have "k>x" by simp
- with C show False by simp
- qed
- ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
- } ultimately show ?thesis using less_linear by blast
-qed
+lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"
+ by (induct t) auto
+lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
+lemma map_tree_greater: "tree_greater k (map f t) = tree_greater k t" by (induct t) simp+
+lemma map_tree_less: "tree_less k (map f t) = tree_less k t" by (induct t) simp+
+lemma map_sorted: "sorted (map f t) = sorted t" by (induct t) (simp add: map_tree_less map_tree_greater)+
+lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
+lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
+lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
+theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t"
+unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_sorted map_color_of)
-lemma map_of_entries:
- shows "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
-proof (induct t)
- case Empty thus ?case by (simp add: RBT.lookup_Empty)
-next
- case (Branch c t1 k v t2)
- hence "map_of (entries (Branch c t1 k v t2)) = RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1" by simp
- also note lookup_entries_aux [OF Branch.prems,symmetric]
- finally show ?case .
-qed
-
-lemma fold_entries_fold:
- "foldwithkey f t x = foldl (\<lambda>x (k,v). f k v x) x (entries t)"
-by (induct t arbitrary: x) auto
-
-lemma entries_pit[simp]: "(k, v) \<in> set (entries t) = entry_in_tree k v t"
+theorem lookup_map [simp]: "lookup (map f t) x = Option.map (f x) (lookup t x)"
by (induct t) auto
-lemma sorted_entries:
- "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
-by (induct t)
- (force simp: sorted_append sorted_Cons tree_ord_props
- dest!: entry_in_tree_keys)+
+
+subsection {* Folding over entries *}
+
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
+ "fold f t s = foldl (\<lambda>s (k, v). f k v s) s (entries t)"
-lemma distinct_entries:
- "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
-by (induct t)
- (force simp: sorted_append sorted_Cons tree_ord_props
- dest!: entry_in_tree_keys)+
+lemma fold_simps [simp, code]:
+ "fold f Empty = id"
+ "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
+ by (simp_all add: fold_def expand_fun_eq)
-hide (open) const Empty insert delete entries lookup map fold union adjust sorted
-
+hide (open) const Empty insert delete entries lookup map_entry map fold union sorted
(*>*)
text {*
@@ -1018,6 +1048,7 @@
used as an efficient representation of finite maps.
*}
+
subsection {* Data type and invariant *}
text {*
@@ -1040,6 +1071,7 @@
$O(\log n)$.
*}
+
subsection {* Operations *}
text {*
@@ -1081,6 +1113,7 @@
@{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
*}
+
subsection {* Map Semantics *}
text {*